We introduce an efficient and robust auto-tuning framework for hyperparameter selection in dimension reduction (DR) algorithms, focusing on large-scale datasets and arbitrary performance metrics. By leveraging Bayesian optimization with a surrogate model, our approach enables efficient hyperparameter selection with multi-objective trade-offs and allows us to perform data-driven sensitivity analysis. By incorporating normalization and subsampling, the proposed framework demonstrates versatility and efficiency, as shown in applications to visualization techniques such as t-SNE and UMAP. We evaluate our results on various synthetic and real-world datasets using multiple quality metrics, providing a robust and efficient solution for hyperparameter selection in DR algorithms.